Three charges, Q1, Q2 and Q3 are located on a straight line. The charge Q3 is located 0.136 m to the right of Q2. The charges Q1 = 1.22 μC and Q2 = -2.46 μC are fixed at their positions, distance 0.301 m apart, and the charge Q3 = 3.03 μC could be moved along the line. For what position of Q3 relative to Q1 is the net force on Q3 due to Q1 and Q2 zero? Give your answer in meters, and use the plus sign for Q3 to the right of Q1.
To find the position of Q3 relative to Q1 where the net force on Q3 due to Q1 and Q2 is zero, we need to calculate the net force on Q3 at different positions and find where it becomes zero.
The force between two charges is given by Coulomb's law:
F = k * |Q1 * Q2| / r^2
Where:
F is the force between the charges,
k is the electrostatic constant (k = 8.99 x 10^9 N*m^2/C^2),
Q1 and Q2 are the charges, and
r is the distance between the charges.
First, let's calculate the force on Q3 due to Q1. Since Q1 and Q3 are both positive charges, the force between them will be repulsive. The force is given by:
F1 = k * |Q1 * Q3| / r1^2
Where:
F1 is the force on Q3 due to Q1,
r1 is the distance between Q1 and Q3.
Next, let's calculate the force on Q3 due to Q2. Since Q2 and Q3 have opposite charges, the force between them will be attractive. The force is given by:
F2 = k * |Q2 * Q3| / r2^2
Where:
F2 is the force on Q3 due to Q2,
r2 is the distance between Q2 and Q3.
The net force on Q3 is given by the vector sum of F1 and F2:
Net Force = F1 + F2
Since we want the net force to be zero, we set the equation equal to zero:
0 = F1 + F2
Now, let's substitute the known values into the equation and solve for r2, the position of Q3 relative to Q1:
F1 = k * |Q1 * Q3| / r1^2
F2 = k * |Q2 * Q3| / r2^2
0 = k * |Q1 * Q3| / r1^2 + k * |Q2 * Q3| / r2^2
We are given the following values:
Q1 = 1.22 μC = 1.22 x 10^-6 C
Q2 = -2.46 μC = -2.46 x 10^-6 C
Q3 = 3.03 μC = 3.03 x 10^-6 C
r1 = 0.301 m
r2 = 0.136 m
Plugging in these values:
0 = (8.99 x 10^9 N*m^2/C^2) * |(1.22 x 10^-6 C) * (3.03 x 10^-6 C)| / (0.301 m)^2 + (8.99 x 10^9 N*m^2/C^2) * |(-2.46 x 10^-6 C) * (3.03 x 10^-6 C)| / (r2)^2
Simplifying the equation:
0 = (34.645 x 10^-3 N*m^2/C^2) / (0.090601 m^2) - (22.195 x 10^-3 N*m^2/C^2) / (r2)^2
To solve for r2, we rearrange the equation:
(22.195 x 10^-3 N*m^2/C^2) / (r2)^2 = (34.645 x 10^-3 N*m^2/C^2) / (0.090601 m^2)
Cross-multiplying:
(22.195 x 10^-3 N*m^2) * (0.090601 m^2) = (34.645 x 10^-3 N*m^2) * (r2)^2
Dividing both sides by (34.645 x 10^-3 N*m^2) and taking the square root:
sqrt((22.195 x 10^-3 N*m^2) * (0.090601 m^2) / (34.645 x 10^-3 N*m^2)) = r2
Simplifying the expression:
r2 = sqrt(0.0572266) = 0.239 m
Therefore, the position of Q3 relative to Q1 where the net force on Q3 is zero is 0.239 meters to the right of Q1.